Double Potential Well: Wave Function & Forces

[michael879]

I remember this homework assignment I did a while back, where we "found" the wave function of a particle in a double potential well. The distance between them was variable and we found that as the distance decreased, the energy of the system went down. This would suggest a force pulling the two wells together right? I know this doesn't explain repelling forces at all, but wouldn't this "work" as an explanation of a force like gravity? treat composite particles as potential wells, and quarks as particles with a wave function that extends to all the other wells (its incredibly small everywhere but the particle its in). Particles made of quarks would feel a weak attractive force to each other.

I doubt this actually works to explain gravity, and maybe the quark example is ridiculous since their wave function is pretty confined. However, anything that acts a well would have this attractive force wouldn't it?

 

[per.sundqvist]

No it's a question of overlap of wave functions in the barrier region, which becomes significant if the distance between the wells decreases (or the barrier is low).

\Psi(x)=c_1\Psi_1(x)+c_2\Psi_2(x),

where Psi_1 and Psi_2 are local (approximative) solutions to the "uncoupled" problem, but with some decay into the barrier (like gaussian, for instance).

The penetration of the wave function into the finite barrier add an extra energy:

\Delta E =V_{barr}\int_0^L \Psi^2(x)\;dx,

where L is the thickness of the barrier.

The electron does not interact with itself! There is no such term in the Hamiltonian. If you have many electrons you would have to include Coulomb interaction of course.

/Per